Beam-shaping optical element and method and program for designing the same

ABSTRACT

A beam-shaping optical element employing an aspherical profileis presented to minimize aberration. A beam-shaping optical element according to the present invention includes an entrance surface and the exit surface, both having a non-circular cross-section in any plane comprising the optical axis. A beam-shaping optical element according to one embodiment of the present invention, has the optical axis coinciding with the Z-axis of a three-axis rectangular XYZ system of coordinates, and the entrance surface and/or the exit surface represented by a mathematical equation comprising a term representing a non-rotationally symmetric aspherical profile, at least one correction term comprising a function of variable X alone and at least one correction term comprising a function of variable Y alone.

FIELD OF THE INVENTION

The present invention relates to a beam-shaping optical element used inoptical pick-up systems for optical storage applications like compactdiscs (CD), digital versatile discs (DVD) or the like, in opticalcommunication systems and in other fields. The present invention has awide range of beam-shaping applications, including those for coupling ofa beam to an optical fiber and those in laser beam printers, scanners,apparatuses for laser machining, high efficient LD optical systems fordiode-pumped solid-state lasers and other apparatuses employing opticalsystems.

BACKGROUND OF THE IVENTION

FIG. 5 shows a schematic layout of an optical pick-up system. Lightemitted by a semiconductor laser 1 is converted into a parallel beam bycollimator 2. The light is then reflected by a folding mirror 3 and ledto objective lens 4, which focuses the light onto an opticalinformation-recording medium 5 for recording and reproducinginformation.

The field pattern of the light emitted by the semiconductor laser 1 iselliptical as shown in FIG. 7, where the major axis is perpendicular tothe layer direction (face of junction). In other words, the light beamhas its energy elliptically distributed in a cross-section of the beam.In case of a usual semiconductor laser for recording applications, theFWHM (full width at half maximum) is approximately 8.5 degrees in thelayer direction and approximately 17 degrees in the directionperpendicular to the layer direction. In the direction perpendicular tothe layer direction (face of junction), the outer of the light from thesemiconductor laser 1 may not couple to the collimator lens 2 and is notconverted into a parallel beam, thus resulting in loss of light into theoptics. Further, the cross-section of the light converted into aparallel beam has an elliptical energy distribution. If the light isfocused onto the surface of the optical information-recording medium 5as an optical spot, the optical spot is also elliptical.

Accordingly, various techniques have been developed so far to shape thelight emitted by semiconductor lasers, with an elliptical energydistribution into one with a substantially circular energy distribution.

FIG. 6 shows an optical system in which prisms are combined to shapelight with an elliptical cross-section into one with a substantiallycircular cross-section. The optical system using prisms has thedrawbacks that it is large in size, expensive and troublesome inassembling operation. Further, the system may produce additionalaberrations. A parallel beam is necessary for the operation of theprisms and a large numerical aperture for the collimator is required.

Another technique has been developed, in which a folding mirror is usedfor beam shaping. For example, refer to Japanese unexamined patentpublication No. 9-167375. In this case, the mirror for beam-shaping isarranged behind a collimator lens, so that a distance between the mirrorfor beam-shaping and the semiconductor laser is lager. Accordingly, alarge numerical aperture for the collimator is required.

Still another technique has been developed, in which an aspherical lenshaving different focal lengths in two directions perpendicular to theoptical axis, is used for beam-shaping. Japanese unexamined patentpublications No. 6-274931 and No. 6-294940 disclose techniques in whicha toroidal lens is used as the aspherical lens. Further, Japaneseunexamined patent publications No. 2001-6202 and No. 2001-160234disclose techniques in which a lens having an anamorphic surface, to bedescribed below, is used. In any of the above-mentioned techniques, onesurface or both surfaces of the collimator lens are made aspherical fora beam-shaping function.

A toroidal surface as a aspherical surface can be obtained by defining aprofile by Equation (1) shown below and rotating the profile around theaxis parallel to X axis and passing through the point on Z axis, distantby R_(y) from the origin. The shape is spherical in Y-Z plane andaspherical in X-Z plane. $\begin{matrix}{{Z(x)} = {\frac{c_{x}x^{2}}{1 + \sqrt{1 - {( {1 + k} )c_{x}^{2}x^{2}}}} + {\sum\limits_{i = 1}^{m}{A_{i}x^{2\quad i}\quad( {X - {Z\quad{plane}}} )}}}} & {{Eq}.\quad(1)}\end{matrix}$c_(x) is the curvature of a curve in the X-Z plane and R_(y) is a radiusof a curve (circle) in the Y-Z plane. The second and succeeding termsare correction terms representing a deviation from the surfacerepresented by the first term.

An anamorphic surface can be represented by Equation (2) shown below.$\begin{matrix}{Z = {\frac{{c_{x}x^{2}} + {c_{y}y^{2}}}{1 + \sqrt{1 - {( {1 + k_{x}} )( {c_{x}^{2}x^{2}} )} - {( {1 + k_{y}} )( {c_{y}^{2}y^{2}} )}}} + {{AR}\lbrack {{( {1 - {AP}} )x^{2}} + {( {1 + {AP}} )y^{2}}} \rbrack}^{2} + {{BR}\lbrack {{( {1 - {Bp}} )x^{2}} + {( {1 + {BP}} )y^{2}}} \rbrack}^{3} + {{CR}\lbrack {{( {1 - {CP}} )x^{2}} + {( {1 + {CP}} )y^{2}}} \rbrack}^{4} + {{DR}\lbrack {{( {1 - {DP}} )x^{2}} + {( {1 + {DP}} )y^{2}}} \rbrack}^{5}}} & {{Eq}.\quad(2)}\end{matrix}$where c_(x) is the curvature of a curve in the X-Z plane and equals1/R_(x) and c_(y) is curvature of a curve in the Y-Z plane and equals1/R_(y). The second and succeeding terms are correction termsrepresenting a deviation from the surface represented by the first term.AR, BR, CR, DR, AP, BP, CP and DP are correction coefficients(constants).

In an optical pick-up system in compact discs (CD), digital versatilediscs (DVD) or the like, the aberrations must be minimized for accurateand high-speed recording and reproducing. Accordingly, the aberrationsfor the above-mentioned lenses having a beam-shaping function must alsobe minimized.

Further, in optical communication systems using semiconductor lasers, asimilar beam-shaping element is required for efficiently coupling beamsemitted by a semiconductor laser to an optical fiber, for example,described in Japanese laying-open of unexamined application (KOKAI) No.11-218649.

However, conventional beam-shaping optical elements using theabove-mentioned equations (1) and (2) for the aspherical surfaces do notalways lead to satisfactory results for minimization of aberrations.

Accordingly, there is a need to use a different description of the lenssurface in order to minimize the aberrations of a beam-shaping opticalelement using an aspherical surface.

SUMMARY OF THE INVENTION

In a beam-shaping optical element according to the present invention,both the entrance surface and the exit surface have a non-circularcross-section in any plane comprising the optical axis.

In a beam-shaping optical element according to the present invention,having an entrance surface, an exit surface located opposite thereto andan optical axis, the optical axis coincides with the Z-axis of athree-axis rectangular XYZ system of coordinates, and at least one ofthe entrance surface and the exit surface is represented by amathematical equation comprising a term representing a non-rotationallysymmetric aspherical profile and correction terms, each correction termbeing a function of either variable X or Y, at least one of thecorrection terms being a function of variable X and at least one of thecorrection terms being a function of variable Y. Hence, the equationdoes not comprise a correction term being a function of both variable Xand variable Y and it does comprise at least one correction termcomprising a function of variable X alone and at least one correctionterm comprising a function of variable Y alone. The beam shaping elementmay have one of the surfaces or both surfaces designed according to themathematical equation.

Accordingly, correction in X direction and that in Y direction can bemade independently over the aspherical profile, permitting moreflexibility in designing the element.

In a special embodiment the beam-shaping optical element at least one ofthe entrance surface and the exit surface has a non-circularcross-section in any plane comprising the optical axis.

In a beam-shaping optical element according to one embodiment of thepresent invention, the at least one correction term comprising afunction of variable X alone comprises a power of X, multiplied by acorrection factor and the at least one correction term comprising afunction of variable Y alone comprises a power of Y, multiplied by acorrection factor. Accordingly, correction in the X direction and thatin the Y direction can be made independently over the asphericalprofile, permitting easier correcting operations.

In a special embodiment the correction factors of the correction termshaving the same powers of X and Y are the same.

In a beam-shaping optical element according to another embodiment of thepresent invention, at least one of the entrance surface and the exitsurface is represented by the equation $\begin{matrix}{Z = {\frac{{c_{x}x^{2}} + {c_{y}y^{2}}}{1 + \sqrt{1 - {( {1 + k_{x}} )( {c_{x}^{2}x^{2}} )} - {( {1 + k_{y}} )( {c_{y}^{2}y^{2}} )}}} + {\sum\limits_{i = 1}^{m}{A_{i}x^{2\quad i}}} + {\sum\limits_{i = 1}^{m}{B_{i}y^{2\quad i}}}}} & {{Eq}.\quad(3)}\end{matrix}$in which c_(x) and c_(y) are the curvature of the surface in thedirection of the X axis and Y axis, respectively, and k_(x), k_(y) andthe correction factor A_(i) and B_(i) are constants. By adjustingfactors in the first term, expressing the aspherical profile,beam-shaping function can be implemented. Further, by independentlychanging the factors of the X and Y correction terms, the aberrations ofthe light beam exiting the exit surface of the beam-shaping element canbe minimized.

In a beam-shaping optical element according to another embodiment of thepresent invention, the values of c_(x) and c_(y) are substantiallydifferent. Accordingly, the beam-shaping optical element has differentaspherical profiles in the X and Y directions.

In a beam-shaping optical element according to another embodiment of thepresent invention, A_(i) is non-zero for at least one value of i andB_(j) is non-zero for at least one value of j. Correction in the Xdirection and correction in the Y direction can be made independently.

In a beam-shaping optical element according to another embodiment of thepresent invention, at least one of the entrance and exit surfaces has ashape for minimizing the wave front aberrations of a light beam from aradiation source having passed through the beam-shaping optical element.Correction in the X direction and correction in the Y direction can bemade independently over the aspherical profile, so that aberrations canbe further reduced in comparison with those caused by conventionalelements.

In a beam-shaping optical element according to another embodiment of thepresent invention, an elliptical cross-section of a beam from asemiconductor laser is converted into an almost circular cross-section.

Accordingly, when the element is used in an optical pick-up system, amore efficient transfer of energy from the semiconductor laser to therecording medium, permitting higher-speed recording and reproducing ofthe optical information-recording medium.

A beam-shaping optical element according to another embodiment of thepresent invention, is positioned between a semiconductor laser and anoptical element for converting light from the semiconductor laser intoparallel, diverging or converging light. Accordingly, the beam-shapingoptical element can be advantageously positioned near the semiconductorlaser for higher energy efficiency and better correction of aberration.

In a beam-shaping optical element according to another embodiment of thepresent invention, a distance from the emitting point of a semiconductorlaser to the entrance surface of the element is smaller than a distancefrom the image of the emitting point formed by the beam-shaping opticalelement to the entrance surface and with the image located in the objectspace.

In a beam-shaping optical element according to another embodiment of thepresent invention, the mathematical equation(NA_(out)/2)(1/NA_(inx)+1/NA_(iny))<1is satisfied, where NA_(out) is a numerical aperture at the exit surfaceand NA_(inx) and NA_(iny) are numerical apertures at the entrancesurface in the X-Z plane and Y-Z plane, respectively.

When the above conditions are satisfied, the beam-shaping opticalelement is provided with a pre-collimator function which decreases adifference in entry angle of the light on the mirror of such abeam-splitter or the like. Incorporating the pre-collimator functioninto the beam-shaping optical element reduces the number of componentsin the optical path. An advantage of splitting the collimation functionover two elements, a pre-collimator and a post-collimator, has theadvantage that the pre-collimator can be integrated with the beam-shaperto a combined element having a good thermal stability. If the entirecollimation function is built into the beam-shaper, the greater opticalstrength of the element reduces its thermal stability. The splitting ofthe collimation function has the additional advantage that thepost-collimator can have a relatively large focal distance, allowing anincreased distance between the beam-shaper having the pre-collimatorfunction and the post-collimator. The increased distance can be used forarranging a coupling element between the beam-shaper and thepost-collimator for coupling in radiation from a further light source orcoupling out light returning from the recording medium to a detectionsystem.

Such an optical pick-up system can advantageously be used in an opticalscanning device for scanning an optical recording medium. The pick-upsystem includes a photo detector for converting light from the opticalrecording medium to an electric signal representing information storedon the record carrier. The the scanning device includes anerror-correction circuit connected to the electric signal. The circuitis arranged for correcting errors in the electric signal from the photodetector.

A method for designing a beam-shaping optical element according to thepresent invention, is directed to a beam-shaping optical element whereinthe optical axis of the beam-shaping optical element coincides with theZ-axis of a three-axis rectangular XYZ system of coordinates, and atleast one of the entrance surface and the exit surface of thebeam-shaping optical element is represented by a mathematical equationcomprising a term representing a non-rotationally symmetric asphericalprofile and correction terms, each correction term being a function ofeither variable X or Y, at least one of the correction terms being afunction of variable X and at least one of the correction terms being afunction of variable Y. In the method for designing a beam-shapingoptical element according to the present invention, designing isperformed in such a way that aberrations are minimized The method fordesigning a beam-shaping optical element according to the presentinvention, comprises the steps of determining constraints including avergence of the beam at the entrance surface and a vergence of the beamat the exit surface and obtaining a merit function for at least wavefront aberrations. The method for designing a beam-shaping opticalelement according to the present invention, comprises the steps ofobtaining a value for the merit function under the above constraints,determining whether or not the value of the merit function reaches adesired value, and adjusting the value of at least one correction termto cause the merit function to approach the desired value.

Accordingly, a correction term comprising a function of variable X aloneand a correction term comprising a function of variable Y alone can beadjusted independently, so that aberrations can be further reduced incomparison with those caused by conventional elements.

A computer program for designing a beam-shaping optical elementaccording to the present invention, is directed to a beam-shapingoptical element wherein the optical axis of the beam-shaping opticalelement coincides with the Z-axis of a three-axis rectangular XYZ systemof coordinates, and at least one of the entrance surface and the exitsurface of the beam-shaping optical element is represented by amathematical equation comprising a term representing a non-rotationallysymmetric aspherical profile and correction terms, each correction termbeing a function of either variable X or Y, at least one of thecorrection terms being a function of variable X and at least one of thecorrection terms being a function of variable Y. By the method fordesigning a beam-shaping optical element according to the presentinvention, designing is performed in such a way that aberrations areminimized. The computer program for designing a beam-shaping opticalelement according to the present invention, has a computer perform thesteps of determining constraints including a vergence of the beam at theentrance surface and a vergence of the beam at the exit surface andobtaining a merit function for at least wave front aberration. Thecomputer program for designing a beam-shaping optical element accordingto the present invention, has a computer perform the steps of obtaininga value for the merit function under the above constraints, determiningwhether or not the value of the merit function reaches a desired value,and adjusting the value of at least one correction term to cause themerit function to approach the desired value.

Accordingly, the at least one correction term comprising a function ofvariable X alone and the at least one correction term comprising afunction of variable Y alone can be adjusted independently, so thataberrations can be further reduced in comparison with those caused byconventional elements.

In a design method and a design program according to one embodiment ofthe present invention, the at least one correction term comprising afunction of variable X alone comprises a power of X, multiplied by acorrection factor and the at least one correction term comprising afunction of variable Y alone comprises a power of Y, multiplied by acorrection factor. Accordingly, correction in X direction and that in Ydirection can be made independently over the aspherical profile,permitting easier correcting operations.

A design method and a design program according to another embodiment ofthe present invention, are directed to a beam-shaping optical element inwhich at least one of the entrance surface and the exit surface isrepresented by the equation $\begin{matrix}{Z = {\frac{{c_{x}x^{2}} + {c_{y}y^{2}}}{1 + \sqrt{1 - {( {1 + k_{x}} )( {c_{x}^{2}x^{2}} )} - {( {1 + k_{y}} )( {c_{y}^{2}y^{2}} )}}} + {\sum\limits_{i = 1}^{m}{A_{i}x^{2\quad i}}} + {\sum\limits_{i = 1}^{m}{B_{i}y^{2\quad i}}}}} & {{Eq}.\quad(3)}\end{matrix}$in which c_(x) and c_(y) are the curvature of the surface in thedirection of the X axis and Y axis, respectively, and k_(x), k_(y) andthe correction factor A_(i) and B_(i) are constants. Accordingly, byadjusting factors in the first term, expressing the aspherical profile,a beam-shaping function can be implemented. Further, by independentlychanging factors of the X and Y correction terms, a function ofminimizing wave aberration and the like can be implemented.

A method for making a beam-shaping optical element according to thepresent invention utilizes the method of designing the beam-shapingoptical element according to the present invention. Accordingly, abeam-shaping optical element with smaller aberrations than those ofconventional beam-shaping optical elements can be made.

A computer program product according to the present inventionis usedwith a computer to implement the design method according to the presentinvention. Accordingly, using the computer program product according tothe present invention, a beam-shaping optical element with smalleraberrations than those of conventional beam-shaping optical elements,can be designed.

DESCRIPTION OF THE DRAWINGS

FIG. 1 shows a beam-shaping optical element according to one embodimentof the present invention;

FIG. 2 shows an optical pick-up system comprising a beam-shaping opticalelement according to one embodiment of the present invention;

FIG. 3 shows an optical pick-up system comprising a beam splittercovered with coating;

FIG. 4 shows a beam splitter covered with coating;

FIG. 5 shows a typical optical pick-up system;

FIG. 6 shows a system in which an elliptical cross-section of a beam isconverted into a circular cross-section through a combination of prisms;

FIG. 7 shows a cross-section of a semiconductor laser;

FIG. 8 shows a relationship between numerical apertures at the entry andexit surfaces of a beam-shaping optical element;

FIG. 9 shows an example of a shape of a surface represented by the firstterm alone of Equation (3);

FIG. 10 shows an example of a shape of a surface represented by Equation(3);

FIG. 11 shows a cross-section of the surface of FIG. 9, cut by a planeof y=0;

FIG. 12 shows a differential curve of the cross-sectional profile ofFIG. 11;

FIG. 13 shows a cross-section of the surface of FIG. 10, cut by a planeof y=0;

FIG. 14 shows a differential curve of the cross-sectional profile ofFIG. 13; and

FIG. 15 shows a method for designing a beam-shaping optical element,according to the present invention.

DETAILED DESCRIPTION

An embodiment of an optical pick-up system using a beam-shaping opticalelement according to the present invention will be described withreference to FIG. 2. A beam-shaping optical element 9 according to thepresent invention is positioned between a semiconductor laser 10 and anelement 11 for converting light from the semiconductor laser 10 intoparallel or converging light (for example, a collimator). An ellipticalcross-section of the beam from the semiconductor laser 10 is shaped intoa substantially circular cross-section when the beam passes thebeam-shaping optical element. The beam-shaping optical element 9delivers light with a diverging angle corresponding to a numericalaperture of the element 11 for converting light into parallel orconverging light. The beam-shaping optical element 9 operates as apre-collimator, the element 11 operates as a post-collimator. The lightreflected on a folding mirror 12 passes through the element 11 forconverting light into parallel or converging light, is focused onto anoptical information-recording medium 14 by an objective system. Theobjective system may have one or more optical elements; the figure showsan objective system having one optical element in the form of lens 13.The beam-shaping optical element is preferably a single element.

In the embodiment shown in FIG. 2, the beam-shaping optical element 9and the element 11 for converting light into parallel or converging one(for example, a collimator lens), are provided separately. The structureof separately providing the elements has the following advantages.First, use of the collimator lens can reduce the change in aberrationdue to movement of the objective lens in a direction perpendicular tothe optical axis. In order to reduce the change in aberration due to themovement of the objective lens, the size of the collimator lens must belarge enough compared to that of the objective lens. Thus, thecollimator lens must be positioned at a predetermined distance from thesemiconductor laser 10. Second, the beam-shaping optical element can beadvantageously positioned near the semiconductor laser, for bettercorrection of aberration, even though the collimator lens must bepositioned at the predetermined distance from the semiconductor laser.

Thus, in the structure of the embodiment, the beam-shaping opticalelement 9 and the element 11 for converting light into parallel orconverging one, are separately provided. However, the present inventioncan similarly be applied to an integral-type element in which abeam-shaping function and a function of converting light into parallelor converging one are provided.

A beam-shaping optical element according to the present invention may beprovided with a pre-collimator function, which reduces a diverging angleof light exiting the element. When such a beam is incident on a beamsplitter arranged after the element, the beam will have smaller anglesof incidence an a mirror coating or anti-reflex coating of the beamsplitter, which simplifies the design of the coatings.

The reason why the optical pick-up system requires a pre-collimatorfunction will be described with reference to FIGS. 3 and 4. In FIG. 3, asemi-transparent folding mirror is used as beam splitter and is arrangedbetween a beam splitter covered with coating and the collimator. Thebeam splitter covered with coating has light either reflect on or passthrough it, depending on the direction of polarization of the light.Light having reflected on the beam splitter covered with coating, thenpasses through the collimator, a ¼ wave length plate and an objectivelens before being focused on an optical information-recording medium.Light having reflected on the optical information-recording medium,passes through the objective lens and then passes through the ¼ wavelength plate, so that the direction of polarization of the light rotatesby 90 degrees. Then the light passes through the beam splitter to bedirected to a photo detector (PD). When light reflects on the beamsplitter on its path towards the optical information-recording mediumand passes through the beam splitter on its path from the opticalinformation-recording medium, characteristics of the beam splittercovered with coating depend on angles of incidence of the light beam onthe beam splitter covered with coating. A large difference in angles ofincidence produces a greater change in phase and a change inreflectivity of the coating. For example, when light is made to reflecton the beam splitter covered with coating as shown in FIG. 4, a largerdiverging angle makes a lager difference in angles of incidence on thebeam splitter covered with coating, between “the upper light” and “lowerlight” from the semiconductor laser. Such a larger difference in anglesof incidence causes a problem that phase and energy of the light are notuniform within the beam spot. In order to avoid this problem, apre-collimator for decreasing a diverging angle, and thus a differencein angles of incidence of the light on the beam splitter covered withcoating, must be positioned between the semiconductor laser and the beamsplitter.

In order to provide a beam-shaping optical element according to thepresent invention with a pre-collimator function, a distance between theemitting point of the semiconductor laser and the entry surface of theelement should be made smaller than a distance between the imaginarypoint of the element and the entry surface of the element. For thispurpose, in designing the element the following condition is added. Morespecifically, in a designing procedure to be described below withreference to FIG. 15, the following condition is added.(NA_(out)/2)(1/NA_(inx)+1/NA_(iny))<1where NA_(out) is a numerical aperture at the exit surface and NA_(inx)and NA_(iny) are numerical apertures at the entrance surface in the Xand Y axes, which are orthogonal to each other and the optical axis. Asshown in FIG. 8, a distance S₁ between the emitting point of thesemiconductor laser and the entry surface of the element is made smallerthan a distance S₂ between the imaginary point of the element and theentry surface of the element.

The spot focused on the optical information-recording medium is arrangedto have a desired size by adjusting a value of numerical aperture of theobjective lens 13, so that recording and reproducing is properlyperformed on the optical information-recording medium 14. Since the beamis shaped in such a way that its cross-section is made substantiallycircular, the spot is also made substantially circular. Further, sincethe cross section of the beam has been made into a substantiallycircular shape before the beam enters the element 11 for convertinglight from the semiconductor laser 10 into parallel or converging light,energy loss of the beam incurred until the beam forms the spot, isminimized. Further, a light beam with minimum aberration is realized.Thus, the resultant higher energy efficiency of the beam enables higherrecording density and higher speed in recording and reproducing of theoptical information-recording medium 14.

The beam-shaping optical element 9 according to the present inventionwill be described below. The beam-shaping optical element 9 according tothe present invention is provided with at least one surface representedby Equation (3) below. Although the first term of Equation (3) isidentical with that of Equation (2), the second and succeedingcorrection terms are different from those of Equation (2). Thecorrection terms of Equation (3) are characterized in that termscomprising X and those comprising Y can be corrected by independentfactors. $\begin{matrix}{Z = {\frac{{c_{x}x^{2}} + {c_{y}y^{2}}}{1 + \sqrt{1 - {( {1 + k_{x}} )( {c_{x}^{2}x^{2}} )} - {( {1 + k_{y}} )( {c_{y}^{2}y^{2}} )}}} + {\sum\limits_{i = 1}^{m}{A_{i}x^{2\quad i}}} + {\sum\limits_{i = 1}^{m}{B_{i}{y^{2\quad i}.}}}}} & {{Eq}.\quad(3)}\end{matrix}$where c_(x) and c_(y) are the curvature of the surface in the directionof the X axis and Y axis, respectively, and k_(x), k_(y) and thecorrection factor A_(i) and B_(i) are constants.

Functions of the correction terms of Equation (3) will be described withreference to FIGS. 9 to 14. FIG. 9 shows an example of a biconic surfacerepresented by the first term alone of Equation (3). FIG. 10 shows asurface represented by Equation (3). In FIG. 9, curvatures in X and Ydirections are different from each other. In FIG. 10, the shape of thesurface is complicated in X direction. FIG. 11 shows a cross-section ofthe surface of FIG. 9, cut by a plane of y=0. Further, FIG. 12 shows acurve representing a differential function of the cross-sectionalprofile of FIG. 11. FIG. 13 shows a cross-section of the surface of FIG.10, cut by a plane of y=0. Further, FIG. 14 shows a differential curveof the cross-sectional profile of FIG. 13.

As seen from the drawings mentioned above, since correction termscomprising X and those comprising Y can be corrected by independentfactors, a surface can be designed with more flexibility. Accordingly,an element generating less aberration while the cross-section of thebeam is made substantially circular, can be designed. On the other hand,since the second and succeeding terms of Equation (2) cannot becorrected by independent factors for X and Y, flexibility in designingis less than in Equation (3).

The outline of the method for designing a beam-shaping optical element,according to the present invention, will be described below withreference to FIG. 15. In designing, a commercial software for simulatingoptical behavior of the beam-shaping optical element (for example, Zemaxfrom Focus Software Inc.) can be used. In step S10, an initial shape ofthe beam-shaping optical element is determined. The initial shape of thebeam-shaping optical element is determined, based on a shape of thecross-section of the beam from the semiconductor laser 10, a numericalaperture of the element 11 for converting light from the semiconductorlaser 10 into parallel or converging light and the like. In step S20,constraints and a merit function are determined. The constraintscomprise states of a beam at the entry and exit surfaces of thebeam-shaping optical element or the like. The merit function is, forexample, that for aberration of the beam. In step S30, a value of themerit function is obtained under the constraints. In step S40 it isdetermined whether or not the value of the merit function has reached adesired value. If the function has reached the desired value, theprocess finishes. If the value has not reached the desired one, one ormore parameters of Equation (3) is adjusted in step S50. According tothe present invention, correction terms comprising X and thosecomprising Y can be corrected by independent factors, so that parameterscan be advantageously adjusted with more flexibility. Further, in stepS60, the constraints are adjusted if necessary. The process goes back tostep S30 and the steps are repeated until the merit function reaches thedesired value.

Further, a program for implementing the above-mentioned designingprocess can be created. The design program may incorporate a program forsimulating optical behavior of the beam-shaping optical element. Thedesign program is arranged to have a computer perform each of the stepsshown in FIG. 15. Further, the design program may be provided with aninteractive function, so that determination of the constraints and themerit function in step S20, adjustment of parameters in Step S50 andadjustment of the constraints in step S60 can be performed throughinteraction with the designer. In this case, a variety of constraintsand merit functions may be stored in a storing device of the computerand displayed for the designer, so that the designer can select any ofthem. Further, ways of adjusting parameters in step S50 and ways ofadjusting constraints in step S60 may be stored in a storing device ofthe computer in various manners and displayed for the designer, so thatthe designer can select any of them.

An example of a shape of the beam-shaping optical element obtained bythe process mentioned above, will be described in Table 1. In theexample, both entry and exit surfaces are represented by Equation (3).Thus, Table 1 shows values of factors of Equation (3). TABLE 1 R_(x)R_(y) k_(x) k_(y) A₄ A₆ B₄ B₆ Surface1(Entry surface) 6.03 −0.60 0.0001.710 −0.0084 −0.0010 1.0289 6.5073 Surface2(Exit surface) −7.60 −2.370.809 0.044 0.0033 −0.0002 0.0006 −0.0007Center thickness; 2.7 mm

Further, a shape in YZ cross-section comprising the optical axis andthat in XZ cross-section comprising the optical axis of the beam-shapingoptical element having surfaces represented by Table 1, are shown inFIG. 1. A shape of the entry surface in YZ cross-section 7 a has anegative curvature, while a shape of the entry surface in XZcross-section 7 b has a positive curvature. Such curvatures permitbeam-shaping from an elliptical cross-section to a substantiallycircular one.

In the example, the beam-shaping optical element is made of olefincopolymer, although it can be made of other plastics.

Aberrations of the beam-shaping optical element of the example asdesigned using Equation (3) are shown in Table 2, in comparison withthose of beam-shaping optical elements having surfaces represented byEquations (1) and (2). The beam-shaping optical elements having surfacesrepresented by Equations (1) and (2) have been designed in a similarprocess with that shown in FIG. 15. TABLE 2 Quadratic λRMS SA 6^(th)SA8^(th)SA AS 4^(th)AS 6^(th)AS AS Total Eq. (3) 0.0000 0.0003 0.00010.0002 0.0000 0.0000 0.0000 0.0004 Eq. (1) 0.0090 0.0001 0.0028 0.00200.0002 0.0005 0.0100 0.0142 Eq. (2) 0.0013 0.0012 0.0006 0.0013 0.00080.0000 0.0004 0.0029

In Table 2, “SA” represents spherical aberration and “AS” representsastigmatic aberration. “Quadratic AS” represents quadratic astigmaticaberration. “Total” represents wave aberration. Any aberration is givenas the root-mean-square value in units of λ. Aberrations caused by thebeam-shaping optical element according to the present invention areremarkably reduced compared with those caused by the beam-shapingoptical elements having surfaces represented by Equations (1) and (2).

The correction terms of Equation (3) permit more flexibility indesigning surfaces, so that a beam-shaping optical element with minimumaberration can be realized.

1. A beam-shaping optical element having an entrance surface, an exitsurface located opposite thereto and an optical axis, wherein theoptical axis coincides with the Z-axis of a three-axis rectangular XYZsystem of coordinates, and at least one of the entrance surface and theexit surface is represented by a mathematical equation comprising a termrepresenting a non-rotationally symmetric aspherical profileandcorrection terms, each correction term being a function of eithervariable X or Y, at least one of the correction terms being a functionof variable X and at least one of the correction terms being a functionof variable Y.
 2. A beam-shaping optical element according to claim 1,wherein the at least one correction term comprising a function ofvariable X alone comprises a power of X, multiplied by a correctionfactor and the at least one correction term comprising a function ofvariable Y alone comprises a power of Y, multiplied by a correctionfactor.
 3. A beam-shaping optical element according to claim 1, whereinthe at least one of the entrance surface and the exit surface isrepresented by the mathematical equation$Z = {\frac{{c_{x}x^{2}} + {c_{y}y^{2}}}{1 + \sqrt{1 - {( {1 + k_{x}} )( {c_{x}^{2}x^{2}} )} - {( {1 + k_{y}} )( {c_{y}^{2}y^{2}} )}}} + {\sum\limits_{i = 1}^{m}{A_{i}x^{2\quad i}}} + {\sum\limits_{i = 1}^{m}{B_{i}y^{2\quad i}}}}$in which c_(x) and c_(y) are the curvature of the surface in thedirection of the X axis and Y axis, respectively, and k_(x), k_(y) andthe correction factor A_(i) and B_(i) are constants.
 4. A beam-shapingoptical element according to claim 3, in which the values of c_(x) andc_(y) are substantially different.
 5. A beam-shaping optical elementaccording to claim 3, in which A_(i) is non-zero for at least one valueof i and B_(j) is non-zero for at least one value of j.
 6. Abeam-shaping optical element according to claim 1, in which at least oneof the entrance and exit surfaces has a shape for minimizing the wavefront aberrations of a light beam from a radiation source having passedthrough the beam-shaping optical element.
 7. A beam-shaping opticalelement according to claim 1, wherein an elliptical cross-section of abeam supplied by a radiation source is converted into a substantiallycircular cross-section.
 8. A beam-shaping optical element according toclaim 1, positioned between a semiconductor laser and an optical elementfor converting a beam from the semiconductor laser into a parallel,diverging or converging light beam.
 9. A beam-shaping optical elementaccording to claim 1, wherein a distance from the emitting point of asemiconductor laser to the entrance surface of the element is smallerthan a distance from the image of the emitting point formed by thebeam-shaping optical element to the entrance surface and with the imagelocated in the object space.
 10. A beam-shaping optical elementaccording to claim 1, wherein the mathematical equation(NA_(out)/2)(1/NA_(inx)+1/NA_(iny))<1 is satisfied, where NA_(out) is anumerical aperture at the exit surface and NA_(inx) and NA_(iny) arenumerical apertures at the entrance surface in the X-Z plane and Y-Zplane, respectively.
 11. An optical pick-up system for scanning anoptical recording medium and provided with a light source and anobjective lens for converging a light beam from the light source on therecording medium, wherein a beam-shaping element according to claim 1 isarranged in the optical path between the light source and the objectivesystem.
 12. An optical scanning device for scanning an optical recordingmedium and provided with an optical pick-up system according to claim11, wherein the pick-up system includes a photo detector for convertinglight from the optical recording medium to an electric signalrepresenting information stored on the record carrier, and the scanningdevice includes an error-correction circuit connected to the electricsignal.
 13. A method for designing a beam-shaping optical element insuch a way that aberrations are minimized, the optical axis of thebeam-shaping optical element coinciding with the Z-axis of a three-axisrectangular XYZ system of coordinates, at least one of the entrancesurface and the exit surface of the beam-shaping optical element beingrepresented by a mathematical equation comprising a term representing anon-rotationally symmetric aspherical profile and correction terms, eachcorrection term being a function of either variable X or Y, at least oneof the correction terms being a function of variable X and at least oneof the correction terms being a function of variable Y, the methodcomprising the steps of: determining constraints including a vergence ofthe beam at the entrance surface and a vergence of the beam at the exitsurface; obtaining a merit function for at least wave front aberration;obtaining a value for the merit function under the above constraints;determining whether or not the value of the merit function reaches adesired value; and adjusting at least one correction term to cause themerit function to approach the desired value.
 14. A method for designinga beam-shaping optical element according to claim. 13, wherein the atleast one of the entrance surface and the exit surface of the element isrepresented by the mathematical equation$Z = {\frac{{c_{x}x^{2}} + {c_{y}y^{2}}}{1 + \sqrt{1 - {( {1 + k_{x}} )( {c_{x}^{2}x^{2}} )} - {( {1 + k_{y}} )( {c_{y}^{2}y^{2}} )}}} + {\sum\limits_{i = 1}^{m}{A_{i}x^{2\quad i}}} + {\sum\limits_{i = 1}^{m}{B_{i}y^{2\quad i}}}}$in which c_(x) and c_(y) are the curvature of the surface in thedirection of the X axis and Y axis, respectively, and k_(x), k_(y) andthe correction factor A_(i) and B_(i) are constants.
 15. A method formaking a beam-shaping optical element comprising the step of designingthe beam-shaping optical element according to claim 13, and the step ofmaking the optical element according to the design.
 16. A computerprogram for designing a beam-shaping optical element in such a way thataberrations are minimized, the optical axis of the beam-shaping opticalelement coinciding with the Z-axis of a three-axis rectangular XYZsystem of coordinates, at least one of the entrance surface and the exitsurface of the beam-shaping optical element being represented by amathematical equation comprising a term representing a non-rotationallysymmetric aspherical profile and correction terms, each correction termbeing a function of either variable X or Y, at least one of thecorrection terms being a function of variable X and at least one of thecorrection terms being a function of variable Y, the program having acomputer perform the steps of: determining constraints including avergence of the beam at the entrance surface and a vergence of the beamat the exit surface; obtaining a merit function for at least wave frontaberration; obtaining a value for the merit function under the aboveconstraints; determining whether or not the value of the merit functionreaches a desired value; and adjusting at least one correction term tocause the merit function to approach the desired value.
 17. A computerprogram for designing a beam-shaping optical element according to claim16, wherein the at least one correction term comprising a function ofvariable X alone comprises a power of X, multiplied by a correctionfactor and the at least one correction term comprising a function ofvariable Y alone comprises a power of Y, multiplied by a correctionfactor.
 18. A computer program for designing a beam-shaping opticalelement according to claim 16, wherein the at least one of the entrancesurface and the exit surface of the element is represented by theequation$Z = {\frac{{c_{x}x^{2}} + {c_{y}y^{2}}}{1 + \sqrt{1 - {( {1 + k_{x}} )( {c_{x}^{2}x^{2}} )} - {( {1 + k_{y}} )( {c_{y}^{2}y^{2}} )}}} + {\sum\limits_{i = 1}^{m}{A_{i}x^{2\quad i}}} + {\sum\limits_{i = 1}^{m}{B_{i}y^{2\quad i}}}}$in which c_(x) and c_(y) are the curvature of the surface in thedirection of the X axis and Y axis, respectively, and k_(x), k_(y) andthe correction factor A_(i) and B_(i) are constants.
 19. A computerprogram product contained in a tangible medium for operating with acomputer in implementing a method as claimed in claim 13.